aiqbench analysis
This paper establishes high-probability convergence guarantees for standard gradient clipping in language model training, specifically addressing heavy-tailed noise and $(L_0,L_1)$-smoothness conditions. By resolving the exponential dependence on smoothness constants found in prior work, the authors provide optimal rates for convex objectives that require no extra oracle calls or function value evaluations. These theoretical advances offer a rigorous foundation for understanding why clipping stabilizes training dynamics in large-scale deep learning systems.
High-Probability Convergence of Clipped SGD under Heavy-Tailed Noise and $(L_0,L_1)$-Smoothness
arXiv:2505.20817v3 Announce Type: replace-cross Abstract: Gradient clipping is widely used in language-model training to control heavy-tailed gradient noise and can improve convergence guarantees over stochastic gradient descent (SGD) under $(L_0,L_1)$-smoothness. Under these joint conditions, a central challenge is to obtain high-probability guarantees without exponential dependence on $L_1R_0$, where $R_0$ bounds the initial distance to a minimizer. We resolve this challenge for convex objectives, establishing, to the best of our knowledge, the first such guarantees for standard Clip-SGD. We assume unbiased stochastic gradients with bounded central $\alpha$-th moment, $\alpha\in(1,2]$. Our bounds have only polylogarithmic dependence on the inverse failure probability and recover known deterministic generalized-smoothness rates when the noise vanishes, as well as the standard high-probability rate under heavy-tailed noise in the classical $L$-smooth setting. The convex rate is attained by a computable output that averages iterates whose sampled stochastic gradients are not clipped, requiring neither function values nor extra oracle calls. We establish a lower bound for Clip-SGD with any fixed stepsize and clipping level, showing that, under the stated iteration conditions, our convex stochastic rate is optimal up to logarithmic factors in the iteration budget. Our upper-bound analysis uses a directional clipping-bias bound to absorb part of the bias into the progress generated by the clipped population gradient, avoiding an exponentially large local smoothness constant. We also obtain nonconvex high-probability guarantees that recover the noiseless rate, match the classical-smooth stochastic iteration rate when $L_1=0$, and have no explicit $L_1$ dependence in the asymptotically dominant stochastic term.
Abstract:Gradient clipping is widely used in language-model training to control heavy-tailed gradient noise and can improve convergence guarantees over stochastic gradient descent (SGD) under $(L_0,L_1)$-smoothness. Under these joint conditions, a central challenge is to obtain high-probability guarantees without exponential dependence on $L_1R_0$, where $R_0$ bounds the initial distance to a minimizer. We resolve this challenge for convex objectives, establishing, to the best of our knowledge, the first such guarantees for standard Clip-SGD. We assume unbiased stochastic gradients with bounded central $\alpha$-th moment, $\alpha\in(1,2]$. Our bounds have only polylogarithmic dependence on the inverse failure probability and recover known deterministic generalized-smoothness rates when the noise vanishes, as well as the standard high-probability rate under heavy-tailed noise in the classical $L$-smooth setting. The convex rate is attained by a computable output that averages iterates whose sampled stochastic gradients are not clipped, requiring neither function values nor extra oracle calls. We establish a lower bound for Clip-SGD with any fixed stepsize and clipping level, showing that, under the stated iteration conditions, our convex stochastic rate is optimal up to logarithmic factors in the iteration budget. Our upper-bound analysis uses a directional clipping-bias bound to absorb part of the bias into the progress generated by the clipped population gradient, avoiding an exponentially large local smoothness constant. We also obtain nonconvex high-probability guarantees that recover the noiseless rate, match the classical-smooth stochastic iteration rate when $L_1=0$, and have no explicit $L_1$ dependence in the asymptotically dominant stochastic term.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2505.20817 [math.OC]
(or arXiv:2505.20817v3 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2505.20817 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 338 words
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